Algebra Examples

$\frac{x^{2} + 14 x + 49}{{(x + 7)}^{3} \sqrt{x + 7}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{x^{2} + 14 x + 7^{2}}{{(x + 7)}^{3} \sqrt{x + 7}}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 1.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 7 + 7^{2}}{{(x + 7)}^{3} \sqrt{x + 7}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

$\frac{{(x + 7)}^{2}}{{(x + 7)}^{3} \sqrt{x + 7}}$

Step 2

Cancel the common factor of ${(x + 7)}^{2}$ and ${(x + 7)}^{3}$ .

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Step 2.1

Multiply by $1$ .

$\frac{{(x + 7)}^{2} \cdot 1}{{(x + 7)}^{3} \sqrt{x + 7}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor ${(x + 7)}^{2}$ out of ${(x + 7)}^{3} \sqrt{x + 7}$ .

$\frac{{(x + 7)}^{2} \cdot 1}{{(x + 7)}^{2} ((x + 7) \sqrt{x + 7})}$

Step 2.2.2

Cancel the common factor.

$\frac{{(x + 7)}^{2} \cdot 1}{{(x + 7)}^{2} ((x + 7) \sqrt{x + 7})}$

Step 2.2.3

Rewrite the expression.

$\frac{1}{(x + 7) \sqrt{x + 7}}$

Step 3

Multiply $\frac{1}{(x + 7) \sqrt{x + 7}}$ by $\frac{\sqrt{x + 7}}{\sqrt{x + 7}}$ .

$\frac{1}{(x + 7) \sqrt{x + 7}} \cdot \frac{\sqrt{x + 7}}{\sqrt{x + 7}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{(x + 7) \sqrt{x + 7}}$ by $\frac{\sqrt{x + 7}}{\sqrt{x + 7}}$ .

$\frac{\sqrt{x + 7}}{(x + 7) \sqrt{x + 7} \sqrt{x + 7}}$

Step 4.2

Move $\sqrt{x + 7}$ .

$\frac{\sqrt{x + 7}}{(x + 7) (\sqrt{x + 7} \sqrt{x + 7})}$

Step 4.3

Raise $\sqrt{x + 7}$ to the power of $1$ .

$\frac{\sqrt{x + 7}}{(x + 7) ({\sqrt{x + 7}}^{1} \sqrt{x + 7})}$

Step 4.4

Raise $\sqrt{x + 7}$ to the power of $1$ .

$\frac{\sqrt{x + 7}}{(x + 7) ({\sqrt{x + 7}}^{1} {\sqrt{x + 7}}^{1})}$

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{x + 7}}{(x + 7) {\sqrt{x + 7}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{\sqrt{x + 7}}{(x + 7) {\sqrt{x + 7}}^{2}}$

Step 4.7

Rewrite ${\sqrt{x + 7}}^{2}$ as $x + 7$ .

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Step 4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

$\frac{\sqrt{x + 7}}{(x + 7) {({(x + 7)}^{\frac{1}{2}})}^{2}}$

Step 4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{x + 7}}{(x + 7) {(x + 7)}^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{x + 7}}{(x + 7) {(x + 7)}^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$\frac{\sqrt{x + 7}}{(x + 7) {(x + 7)}^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$\frac{\sqrt{x + 7}}{(x + 7) {(x + 7)}^{1}}$

Step 4.7.5

Simplify.

$\frac{\sqrt{x + 7}}{(x + 7) (x + 7)}$

Step 5

Simplify the denominator.

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Step 5.1

Raise $x + 7$ to the power of $1$ .

$\frac{\sqrt{x + 7}}{{(x + 7)}^{1} (x + 7)}$

Step 5.2

Raise $x + 7$ to the power of $1$ .

$\frac{\sqrt{x + 7}}{{(x + 7)}^{1} {(x + 7)}^{1}}$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{x + 7}}{{(x + 7)}^{1 + 1}}$

Step 5.4

Add $1$ and $1$ .

$\frac{\sqrt{x + 7}}{{(x + 7)}^{2}}$